my $str = "/One/Two/Three";

# Start with the whole thing in the array.  You could
# start with @paths = () if you'd rather not include
# the current directory/path/category/whatever.
my @paths = ($str);

while(   # Spreading this out so I can comment it...

$str =~ s[
 ^             # Match from the start of the string...
  (            # Save this in $1
   /           # The first /
   .+          # Followed by one or more <anything>'s
               #  (including /'s!!), all the way up to
               #  the last / (see below)
  )            # Don't save anything else in $1
  /            # The very last / in the string.  This
               #  is what causes the above .+ to stop
               #  before the end of the string.
  [^/]*        # We want an optional word after
               #  the last / but NOT /'s (otherwise
               #  it wouldn't be the last one, would it?)
 $             # Make sure don't stop before the end
               #  of what's left of the string.

][$1]x # Replace it all with $1, which contains
       # everything up to (but not including)
       # the last /.  Note the x option, let's us
       # include comments and space it out.
) {
    # Push the current version of $str onto the list.
    # It's whatever we have left after chopping off
    # the last word and it's /.
    push @paths, $str;
}